I set a couple of squads of fighters to "night", in the hopes of intercepting partison resupply. However, I never have observed seen a battle result suggesting that I have been successful. Is this not possible?

Also, in real life, night fighters were generally specially equipped aircraft (Mostly Me 110 and Me 410 IIRC). Does the success of night missions have anything to do with aircraft type?

The Soviet night supply missions are not reported by battle icons for the Axis player. This is to keep the locations of the partisan units secret, until the become active and start destroying rail lines. The only way to know whether your night fighters are doing anything is to check the number of air-to-air kills on the air groups each turn. This is tedious, but it will give you an idea of whether you're doing anything.